Quadratic head loss approximations for optimisation problems in water supply networks

Edo Abraham,Filippo Pecci,Ivan Stoianov

doi:10.2166/hydro.2017.080

Edo Abraham, Filippo Pecci + Show 1 more

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https://doi.org/10.2166/hydro.2017.080

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Abstract

This paper presents a novel analysis of the accuracy of quadratic approximations for the Hazen–Williams (HW) head loss formula, which enables the control of constraint violations in optimisation problems for water supply networks. The two smooth polynomial approximations considered here minimise the absolute and relative errors, respectively, from the original non-smooth HW head loss function over a range of flows. Since quadratic approximations are used to formulate head loss constraints for different optimisation problems, we are interested in quantifying and controlling their absolute errors, which affect the degree of constraint violations of feasible candidate solutions. We derive new exact analytical formulae for the absolute errors as a function of the approximation domain, pipe roughness and relative error tolerance. We investigate the efficacy of the proposed quadratic approximations in mathematical optimisation problems for advanced pressure control in an operational water supply network. We propose a strategy on how to choose the approximation domain for each pipe such that the optimisation results are sufficiently close to the exact hydraulically feasible solution space. By using simulations with multiple parameters, the approximation errors are shown to be consistent with our analytical predictions.

Highlights

The optimal management of water supply networks requires the satisfaction of multiple objectives, ranging from leakage reduction to improvements in water quality and system resilience
Quadratic approximations have been effectively used to formulate head loss constraints in different optimisation problems for water supply networks. Quantifying and controlling their absolute errors, which affect the degree of constraint violations of feasible candidate solutions, has a strong impact on the application of mathematical optimisation
Both the HW and DW friction head loss models need smooth approximations to be posed as explicit constraints in mathematical optimisation

Summary

INTRODUCTION

The optimal management of water supply networks requires the satisfaction of multiple objectives, ranging from leakage reduction to improvements in water quality and system resilience. The solution of a wide range of problems for the optimal design and operation of water supply networks (Vairavamoorthy & Lumbers ; Bragalli et al ; Gleixner et al ; Menke et al ) by using mathematical programming methods requires rigorously investigated and formulated smooth approximations of the friction head loss formulae. The results in Eck & Mevissen ( ) showed that in practical applications, the use of polynomial quadratic friction loss formulae does not affect significantly the distribution of network pressures and flows These approximations were not analytically discussed and accuracy bounds were not provided. We look for a quadratic polynomial function hf(q) 1⁄4 aq[2] þ bq close to the HW friction head loss formula over the range of flows [0, Qmax]. We focus on two different approaches and analyse their goodness for optimisation problems in water supply networks

A QUADRATIC MODEL THAT MINIMISES ABSOLUTE ERRORS

A QUADRATIC MODEL THAT MINIMISES RELATIVE ERRORS

NUMERICAL RESULTS AND DISCUSSION

CONCLUSIONS